*Isaac Barrow was the first inventor of the infinitesimal Calculus; Newton got the main idea of it from Barrow by personal communication; and Leibniz also was in some measure indebted to Barrow’s work, obtaining confirmation of his own original ideas, …from the copy of Barrow’s book that he purchased in 1673.*

This opening comment by Child arose from his study of the collected lectures of Isaac Barrow, (*LECTIONES Geometricæ*) published in London in 1670. He goes on to say:

*The above is the ultimate conclusion that I have arrived at, as the result of six months’ close study of a single book, my first essay in historical research.*

Originally published in 1916 by Open Court Publishing, this Dover reprint of Child’s book is a recent addition to their Phoenix series. Well known among historians, it is much quoted in the general mathematical histories, and receives particular mention in those books dealing solely with the historical development of the calculus. But having been out of print for so many years, it is certainly deserving of wider circulation.

Reading the book, one is struck by two remarkable achievements. Firstly, there is the work and life of Isaac Barrow himself and, secondly, there are the accomplishments of J. M. Child, who translated, abridged, and provides commentary upon Barrow’s pioneering work on the calculus. In the process, he has made the work of this 17th century scholar accessible to the majority of us who know no Latin, and who struggle to decipher archaic modes of mathematical expression.

Little is known of J. M. Child, other than the fact that, by 1940, he had written or co-authored various school and university textbooks; that, prior to the first world war, he taught mathematics to Rolls Royce engineers at Derby Technical College; and that he later joined the staff of Manchester University. Yet, in addition to the book under review, he subsequently produced a similarly famous work on The Early Mathematical Manuscripts of Leibniz *.* Such deeds seem all the more creditable when one imagines Child tucked away in the Edwardian industrial town of Derby, teaching engineers, builders, draftsmen, etc. by day, and going home at night to apply his knowledge of Latin to the task of deciphering Barrow’s manuscripts. Moreover, to have carried out this solitary task within a six-month period, and as a first venture into the history of mathematics, is a truly impressive feat.

As for Barrow’s book, why the reference to geometry and no mention of ‘calculus’? There are two reasons for this. Firstly, Barrow was a geometer in the classical mould and, as Lucasian Professor of Mathematics, he delivered a series of fifteen lectures between the years 1663 and 1667 — the protracted time span being due to several outbreaks of the plague, during which Cambridge University was closed for extended periods. Although the style of these lectures was decidedly geometric, their main focus was upon problems of tangency and quadrature, for which Barrow invoked the use of Greek geometrical methods. However, it was to be another twenty years before Leibniz introduced the word ‘calculus’ into mathematics (by means of his first paper on these matters, in 1684).

A simplistic summary of Barrow’s main achievements would be to say that, by means of classical geometry, he solved the problem of tangency, invoked the notion of differential triangle and he showed that tangency (differentiation) and quadrature (integration) are inverse processes. He also had insight into methods of integration on solids of revolution and did some original work on the rectification of curves. On the other hand, he made no use of algebra (which he distrusted) and he did not subscribe to the arithmetization of geometry.

However, it has to be said that Barrow’s methodology was not solely dependent upon Greek geometry because he also employed Galileo’s method of composition of motions to generate curves and, to this end, he also used ‘time’ as an independent variable (rather than as a parameter). Consequently, being a teacher of Isaac Newton, this must have had some bearing upon the latter’s development of the fluxional approach to calculus.

Yet another of the many influences upon Barrow was Cavalieri, whose notion of indivisibles he employed to a greater extent than Archimedes’ method of exhaustion.

And, in his very first lecture, he provides the following charming comment as to his thoughts on such phenomena:

*To every instant of time, or indefinitely small particle of time, (I say instant or indefinite particle, for it makes no difference whether we suppose a line to be composed of points or indefinitely small linelets; and so in the same manner, whether we suppose time to be made up of instants or indefinitely small timelets;….*

In other words, Barrow doesn’t distinguish between lines and indefinitely small rectangles as constituent parts of area.

Thanks to Child, this book is packed with many such interesting observations. This is because Barrow’s work is so uniquely ingenious and also because Child has elicited its most salient features. He also expresses Barrow’s geometrically derived results in the language and symbolism of contemporary mathematics and provides much additional historical commentary in the process. Finally, his brief biographical portrait of Barrow creates the impression of a fascinating character — a man of many parts who was one of the first to give formal recognition of the genius of Isaac Newton. John Aubrey, one of Barrow’s contemporaries, describes him thus:

*He was a strong, and stowt man: and feared not any man.* [*When a schoolboy at Charterhouse*] *he would fight with the Butchers’ boyes in St. Nicholas’ shambles, and he be hard enough for any of them.*

*He was abroad five yeares, viz in Italie, France, Germany and Constantinople. As he went to Constantinople, two men of Warre (Turkish-shipps) attacqued the Vessel wherein he was, in which engagement he shewed much valour in defending the vessel.*

[An extract from the mini-biography of Isaac Barrow in Aubrey's *Brief Lives* .]

Peter Ruane (ruane.p@blueyonder.co.uk) is retired from university teaching, where his interests lay predominantly within the field of mathematics education